Multiscale Computations for Highly Oscillatory Problems

3

1 The University of Texas at Austin, Austin, TX 78712,ariel,engquist,[email protected]

2 Department of Numerical Analysis, KTH, 100 44 Stockholm, Sweden3 Trasko–Storo Institute of Mathematics

Summary. We review a selection of essential techniques for constructing computational mul-tiscale methods for highly oscillatory ODEs. Contrary to the typical approaches that attemptto enlarge the stability region for specialized problems, these lecture notes emphasize howmultiscale properties of highly oscillatory systems can be characterized and approximated ina truly multiscale fashion similar to the settings of averaging and homogenization. Essentialconcepts such as resonance, fast-slow scale interactions, averaging, and techniques for trans-formations to non-stiff forms are discussed in an elementary manner so that the materials canbe easily accessible to beginning graduate students in applied mathematics or computationalsciences.

1 Introduction

Oscillatory systems constitute a broad and active field of scientific computations.One of the typical numerical challenges arises when the frequency of the oscillationsis high compared to either the time or the spatial scale of interest. In this case, the

of sampling oscillations adequately by numerical discretizations over a relativelylarge domain. Several general strategies for dealing with oscillations can be foundin literature, for example, asymptotic analysis [5, 22, 23], averaging [20, 29], enve-

tions, which are not oscillatory in the domain of interests. For example, the centeror frequency of oscillators may vary slowly in time. Indeed, it is often the case thatthe quantities of interest are related to these non-oscillatory structures. Reduction inthe computational costs is thus possible by avoiding direct resolution of the oscilla-

the wave equation of the form A(x,t)exp(S(x,t)/ε) is computed via solutions of aneikonal equation for the phase S and transport equations for the amplitude A. Sinceeikonal and transport equations do not depend on the ε-scale oscillations, the cost of

Multiscale Computations for Highly OscillatoryProblems

lope tracking [27, 28], explicit solutions to nearby oscillatory problems [25, 30, 31].These strategies typically utilize some underlying structures, related to the oscilla-

cost for computations can typically become exceedingly expensive due to the need

tions. Take geometrical optics [13, 21] for instance. The high frequency solution of

Heinz-Otto Kreiss , and Richard Tsai1,2Gil Ariel1,*, Bjorn Engquist ,*,

* Ariel, Engquist, and Tsai are partially supported by NSF grant DMS-0714612.

1,*

238 G. Ariel, B. Engquist, H.-O. Kreiss and R. Tsai

computation is formally independent of the fast scale as well. These current lecturenotes focus on building efficient multiscale numerical methods that only sample thefast oscillations. The sampled information is used to describe an effective time evo-lution for the system at longer time scales. The general approach underlying thesemethods come from the theory of averaging.

In these notes, we consider systems of ordinary differential equations (ODEs)which evolve on two widely separated time scales. Common examples are

1. Perturbed linear oscillations:

εx′ = Ax + εg(x), (1)

where A is diagonalizable and has purely imaginary eigenvalues. The class ofexamples include Newton’s equation of motion for perturbed harmonic oscilla-tors

εx′′ = −Ω 2x + εg(x),

which are found in many applications. Here, the parameter ε , 0 < ε 1, char-acterizes the separation of time scales in the system.

2. Fully nonlinear oscillations induced from dissipation in the systems. Examplesinclude Van der Pol oscillators and other relaxation oscillators.

3. Weakly coupled nonlinear oscillators that are close to a slowly varying periodicorbits. Examples include Van der Pol (with small damping) and Volterra-Lotkaoscillators.

Efficient and accurate computations of oscillatory problems require significantknowledge about the underlying fast oscillations. Using either analytical or numer-ical methods, our general underlying principle is to model oscillations and sampletheir interactions. Very often, analytical methods do not yield explicit solutions, andsuitable numerical methods need to be applied.

One of the current major thrusts is in developing numerical methods which allowlong time computation of oscillatory solutions to Hamiltonian systems. The interestin such systems comes from molecular dynamics which attempts to simulate someunderlying physics on a time scale of interest. These methods typically attempt toapproximately preserve some analytical invariance of the solutions; e.g. the totalenergy of the system, symplectic structures, or the reversibility of the flow. Detailedreviews and further references on this active field of “geometric integration” can befound in [18] and [26].

The Verlet method and other similar geometric integrators are the methods ofchoice for many highly oscillatory simulations. They require, however, time stepsthat are shorter than the oscillatory wavelength ε and therefore cannot be used whenε is very small.

wavelength ε but they apply only to restricted classes of differential equations [18].In a way that resembles the discussion of geometrical optics above since these meth-ods explicitly use the exponential function to represent the leading terms in the os-cillations. They work well for problems that are smooth perturbations of problemswith constant coefficients.

Exponential integrators allow for time steps that are longer than the oscillatory

Multiscale computations for highly oscillatory problems 239

Another general approach for dealing with multiscale phenomena computation-ally can be referred to as boosting [33]. The general idea is to artificially “twig” or“boost” the small parameter ε so that the stiffness of the problem is reduced. Knownmethods that fall into this category are Chorin’s artificial compressibility [7] and theCar-Parrinello method used in molecular dynamics [6].

This tutorial deviates from previous texts in that we do not rely or assume somespecific properties or a special class of ODEs such as harmonic oscillations or Hamil-tonian dynamics. Instead, the multiscale methods discussed here compute the effec-tive behavior of the oscillatory system by integrating the oscillations numerically inshort time windows and sampling their interactions by suitable averaging. Indeed,one of the main goals of this text is to make the ideas discussed above mathemat-ically meaningful. Subsequent sections will define what we mean by the effectivebehavior of a given highly oscillatory system, describe the theory of averaging, thestructure of our multiscale algorithms and its computational complexity.

The objectives of multiscale computations

One of the major challenges in problems involving multiple scales is that an accuratecomputations, attempting to resolve the finest scales involved in the dynamics may becomputationally infeasible. In the classical numerical analysis for ODEs, the impor-tant elements are stability, consistency and ultimately convergence. In the standard

size goes to zero. The errors depend on powers of the eigenvalues of the Jacobianof the ODE’s right hand side and the step size. However, in our multiscale setting,

consider the asymptotic cases when the frequency of the fastest oscillations, which

need to rethink what consistency and convergence means in the multiscale setting.One possibility is the following: let E(t;4,ε) denote the error of the numerical ap-proximation at time t, using step size 4 and for problems with ε−1 oscillations, weconsider the convergence of E for 0 < ε < ε4

lim4−→0

(sup

0<ε<ε4

E(t;4,ε)

).

In other words, with a prescribed error tolerance E , the same step size ∆ can be usedfor small enough ε . While this notion of convergence may not be possible for thesolutions of many problems, we may ask for the convergence of some functions orfunctionals of the solutions. Throughout the notes we discuss results from the pre-spective of a few key questions: what is the motivation for constructing a multiscale

A first example, suggested by Germund Dahlquist, is the drift path of a mechan-ical alarm clock, moving due to fast vibrations when it is set off on a hard surface. Ifthe drift path depends only locally in time on the fast oscillations, then it is reasonable

is proportional to 1/

similar to the high frequency wave propagation or homogenization, we would like to

ε, tends to infinity, before the step size is sent to zero. Hence, we

theory, any stable consistent method converges to the analytical solution as the step

from traditional numerical computations?algorithm? What is being approximated? How does our multiscale approach differ


to design a scheme that evolves the slowly changing averaged drift path by measur-ing the effects of the fast solutions only locally in time. Herein lies the possibility ofreducing the computational complexity.

A second example is Kapitza’s pendulum — a rigid pendulum whose pivot isattached to a strong periodic forcing is vibrating vertically with amplitude ε and fre-quency 1/ε . When the oscillations are sufficiently fast, the pendulum swings slowlyback and forth, pointing upwards, with a slow period that is practically independentof ε . Obviously, in the absence of the oscillatory forcing, the pendulum is only sta-ble pointing downwards. Pyotr Kapitza (physics Nobel Laureate in 1978) used thisexample to illustrate a general stabilization mechanisms [17]. This, and similar sim-ple dynamical systems are often used as example benchmark problems to study howdifferent methods approximates highly oscillatory problems.

The following assumptions are made throughout these notes: in the fastest scale,the given system exhibits oscillations with amplitudes independent of ε , and that at alarger time scale, some slowly changing quantities can be defined by the oscillatorysolutions of the system. To facilitate our discussion, we now present our model sce-nario described by the following two coupled systems. Consider a highly oscillatorysystem in Rd1 coupled with a slow system in Rd2 :

εx′ = f (x,v,t)+ εg(x,v,t), (2)

v′ = h(v,x,t), x(0) = x0 ∈ Rd1 , v(0) = v0 ∈ R

d2 . (3)

We assume that x is highly oscillatory, and v is the slow quantity of interest. However,without proper information about x, v cannot be found. We are also interested in someslowly varying quantity that is being defined along the trajectories of x:

β ′ = ψ(β ,t;x( · )). (4)

In a following section, we shall see that for very special initial conditions, the so-lutions of (2) may be very smooth and exhibit no oscillations. The problem of ini-tialization, i.e. finding the suitable initial data so that the slowly varying solutionscan be computed appear in meteorology. We refer the readers to the paper of Kreissand Lorenz [25] for further reading on the theory for the slow manifolds. However,in many autonomous equations, e.g. linear equations, the only slowly varying solu-tions in the system are the equilibria of the system. For problems like the invertedpendulum, it is clear that the slowly varying solutions are not of interest. Then somecomplicated interactions between the oscillations must take place, and one must lookinto different strategies in order to characterize the effective influence of the oscilla-tions in x in the evolution of v.

Our objective is to accurately compute the slowly changing quantity v in a longtime scale (i.e. 0 ≤ t ≤ T, for some constant T independent of ε). Furthermore, wewish to compute it with a cost that is at least sublinear to (ideally independent of ) thecost for resolving all the fast oscillations in this time scale. In general, our objectivemay be achieved if fast oscillations are computed only in very short time intervals andyet the dynamics for those slowly changing quantities is consistently evolved. Figure1 depicts two possible schematic structures for such an algorithm. In this section, we


give a few examples of where such type of slowly changing quantities occur; theseconsist of systems with resonant modes and weakly perturbed Hamiltonians systemsso that invariances are changing slowly. Furthermore, we shall see in the next sectionthat these slowly changing quantities may be evaluated conveniently by short timeaveraging with suitable kernels.

hx(0)

x

ξ

η

micro−solver

Macro−solverH

δt*

T=tn t +ηn

Fig. 1.

Let us briefly comment on the relation of these notes to the standard stiff ODEsolvers for multiscale problems with transient solutions, [8, 19]. A typical example

possible as in the macro-solver in Fig. 1. Special properties of stiff ODE methods

modes are present for all times and may interact to give contributions to the slowermodes.

1.1 Example oscillatory problems

Linear systems with imaginary eigenvalues

x′ = iλ x, λ ∈ R.

The solution is readily given by x(t) = x(0)eiλ t . Note that this system is equivalentto the system in R2: (

xy

)′=(

0 λ−λ 0

)(xy

).

suppress the fast modes and only the slower modes need to be well approximated.

of such stiff problems is equation (1) where the eigenvalues of A are either negative or

the type of the micro-solver in Fig. 1. After the transient, much longer time steps are

Problems with highly oscillatory solutions are much harder to simulate since the fast

zero. The initial time steps are generally small enough to resolve the transient and of


Hamiltonian systems

Hamiltonian dynamics are defined by the partial derivatives of a Hamiltonian func-tion H(q, p) which represent the total energy of the system. Here, q is a generalizedcoordinate system and p the associated momentum. The equations of motion aregiven by

q′ = Hp(q, p),p′

q (5)

where Hp and Hq denote partial derivatives of H with respect to p and q, respectively.In Hamiltonian mechanics, H(p,q) = 1

2 p2 +V (q) and the dynamics defined in (5)yield Newton’s equation of motion q′′

q

then the solutions of this equation are typically oscillatory. An important class ofequations of this type appear in molecular dynamics with pairwise potentials

H(p,q) =12

N

∑i=1

1mi

pTi pi +

12

N

∑i, j=1

Vi j i j

Notable examples are

Vi j(r) =−Gmim j

r(electric or gravitational potential)

and

Vi j(r) = 4εi j

((σi j

r

)12−(σi j

r

)6)

, (Lennard-Jones potential)

for all i 6= j, etc.

Volterra–Lotka

This is a simplified model for the predator-prey problem in population dynamics.In this model, x denotes the population of a predator species while y denotes thepopulation of a prey species

x′ = x(

1 − yν

), (6)

′ =yν

(x − 1).

An example trajectory is depicted in Fig. 2.

Relaxation oscillators

The Van der Pol oscillator is another typical example of nonlinear oscillators. Oneversion of the equation for a Van der Pol oscillator takes the form

= −∇ V (q). If V (x) is a convex function

where pi and qi are components of the vectors p and q.

y

= −H (q, p),

(|q − q |),


0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

x

y

t=0

t=0.046, 3.373 t=3.284

t=3.31

Fig. 2. The trajectory of the Volterra–Lotka oscillator (6) with ν = 0.01, x(0) = 0.5 and y(0) =1.

x′ = y − (x2 − 1)x,y′ = −x.

resistor, inductor, and a capacitor; the state variable x corresponds to the current inthe inductor and y the voltage in the capacitor. It can be shown that there is a uniqueperiodic solution of this equation and other non-equilibrium solutions approach itas time increases. This periodic solution is called the limit cycle or the invariantmanifold of the system. A general result for detecting periodic solutions for such

periodic trajectory of a solution.

x′ = −1 − x + 8y3, (7)

y′ =1ν(−x + y − y3) ,

where 0 < ν 1. For small ν , trajectories quickly come close to the limit cycledefined by −x + y − y3 = 0. The upper and lower branches of this cubic polynomialare stable up to the turning points at which dx/dy = 0. For any initial condition, thesolution of (7) is rapidly attracted to one of the stable branches on an O(ν) timescale. The trajectory then moves closely along the branch until it becomes unstable.At this point the solution is quickly attracted to the other stable branch. The trajectoryis depicted in Fig. 3.

type of systems on a plane is the Poincare–Bendixson theorem, which says that if a

As a second example, consider [9]

This equation can be interpreted as a model of a basic RLC circuit, consisting of a

compact limit set in the plane contains no equilibria, it is a closed orbit; i.e. it is a


−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−1.5

−1

−0.5

0

0.5

1

1.5

x=y−y3

Fig. 3. The trajectory and slow manifold of the relaxation oscillator (7)

1.2 Invariance

Hamiltonian systems:

is the energy H(p,q)

ddt

H(p(t),q(t)) = Hp p′ + Hqq′ = 0

=⇒ H(p(t),q(t)) = H(p0,q0) = const.

Let us prove Liouville’s theorem on volume preservation of Hamiltonian systems.Consider a smooth Hamiltonian H(p,q). Let

ϕt(p0,q0) =(

p(t; p0,q0)q(t; p0,q0)

).

Hence,

ddt

∂ϕt

∂ (p0,q0)=(

−Hpq Hqq

Hpp Hqp

)( ∂ϕt∂ (p,q)

(p(t),q(t))

), q, p ∈ R

=⇒ ddt

det∂ϕt

∂ (p0,q0)= 0. (8)

Consider t as a parameter for the family of coordinate changes (diffeomorphisms)φt : (p0,q0) 7→ (p(t; p0,q0),q(t, p0,q0)). Then we have the following change of co-ordinates formula, for any fixed t,

∫

Vf (p,q)dqd p =

∫

Uf (φt (p0,q0))Jd p0dq0,

The Hamiltonian equations of motion (5) admit several invariances. First and foremost


where V = φt(U) and

J := det∂φt

∂ (p0,q0).

Thus, (8) implies thatdJdt

≡ 0

In particular, taking f ≡ 1 implies that U and V have the same volume in the phase(p,q)-space.

Volterra-Lotka oscillators

Let I(u,v) = logu−u+2logv−v. Substituting (6) yields (d/dt)I(u(t),v(t)) = 0 fort > 0.

Let u(t) = (cos(t),sin(t)) and v(t) = (cos(t + φ0),sin(t + φ0)) be the solutions ofsome oscillators. Then

ξ (t) = u(t) · v(t) = cosφ0

measures the phase difference between u(t) and v(t) and remains constant in time.In view of the above examples, the following questions naturally appear:

• Can one design numerical schemes so that the important invariances are pre-served?What is the computational cost or benefit?How well do common numerical approximations preserve known invariances ofinterest and for what time scale?

• What is the importance of preserving invariances? How can this notion be quan-tified?

• How do small perturbations affect the invariances? For example, in the followinglinear system which conserves energy for ε = 0: x′′ = −ω2x + ε cos(λ t). Howdoes weak periodic forcing affect the energy? At what time scale does the forcingbecome important? Can these effects be computed efficiently?

1.3 Resonance

Resonances among oscillations appear in many situations. For example, in pushinga child on a playground swing. It is intuitively clear that unless the swing is pushedat a frequency which is close to the natural oscillation frequency of the swing, thechild will be annoyed. However, when the swing is pushed at the right frequency, theamplitude of the swing is gradually increasing. In this subsection, we review a fewbasic examples of resonance.

Relative phase between two linear oscillators

Some aspects of these issues and others are discussed in [18] and [26].


Resonance in a forced linear spring

We start with a linear spring under periodic forcing,

x′′ = −ω2x + cosλ t.

ddt

(xx′

)=(

0 1−ω2 0

)(xx′

)+(

0cosλ t

),

with initial condition

(x0

x′0

). One can show that the solution operator for the homo-

geneous problem is

St =(

cosωt ω−1 sinωt−ω sinωt cosωt

)

so that the solution for the inhomogeneous problem is(

xx′

)= St

(x0

x′0

)+∫ t

0St−s

(0

cosλ s

)ds

=⇒(

xx′

)= St

(x0

x′0

)+∫ t

0

(ω−1 sin(ωt − ωs) cosλ s

cos(ωt − ωs) cosλ s

)ds.

When λ = ω , resonance happens. More precisely, we see that

x(t) = x0 cosωt +x′

0

ωsin ωt +

1ω

∫ t

0sin(ωt − ωs) cosωsds

=⇒ x(t) = x0 cosωt +x′

0

ωsinωt +

t2

sinωt.

In addition,∫ t

0sin(ωt − ωs)cosωsds =

∫ t

0(sinωt cosωs− sinωscosωt)cosωsds

= sin(ωt)∫ t

0cos2(ωs)ds − cos(ωt)

∫ t

0sin(ωs)cos(ωs)ds

= sin(ωt)∫ t

0

12(1 + cos2ωs)ds− cos(ωt)

∫ t 12

sin(2ωs)ds

=t2

sinωt︸︷︷︸result of resonance

+12

∫ t

0

=t2

sinωt︸︷︷︸

+12

∫ t

0sin(ωt − 2ωs)ds

︸︷︷︸.

If λ 2 6= ω2,we have

x =(

x0 −1

ω2 − λ 2

)cosωt +

x′0

ωsin ωt +

1ω2 − λ 2 cosλ t.

Rewriting into a first order system, we obtain

(sin ωt cos(2ωs −cos ωt sin 2ωs ds)( ) ) ( () )

result of resonance

0

= 0


Exercise 1. Compute the solution of the forced oscillation under friction: for µ > 0

x′′ = −2µx′ − ω2x + cosλ t.

Show that if λ = ω , the amplitude of the oscillations remains bounded and is largestwhen λ =

√ω2 − 2µ2.

Resonance in first order systems

Consider,

x′ =iε

Λx + f(

x,tε

), x(0,ε) = x0,

where Λ is a diagonal matrix. We make the substitution:

x(t) = eiε Λt w(t), w(0) = x0,

and obtain the corresponding equation for w:

w′ = e− iε Λt f

(e

iε Λtw,

tε

), w(0) = x0.

The simplest type of resonance can be obtained by taking f (x,t/ε) = x, i.e., w′ = w.We see that the resonance occurs due to the linearity of f in x, and that it results in |x|changing at a rate independent of ε . If f (x,t/ε) = fI(x)+ exp(it/ε) and one of thediagonal elements of Λ is 1, then similar to the resonance in the forced linear spring,the resonance with the forcing term contributes a linear in time growth of |x|.

Resonances may occur due to nonlinear interaction. Following the above exam-ple, take

Λ =( )

and f (x,t) =(

0−x4

1x−12

).

Hence,

x =(

eit/ε w1

e2it/ε w2

)=⇒

w′ =(

e−it/ε

e−2it/ε

)(0

−e4it/ε e−2it/ε w41w−1

2

)=(

0−w4

1w−12

).

Again, due to the resonance in the system, |x2| is changing at a rate that is indepen-dent of ε .

2 Slowly varying functions of the solutions

In this section we shall study the effect of non-linear interactions. We excerpt impor-tant results from [24] and [1, 2, 14].

1 00 2

00


2.1 Problems with dominant fast linear oscillations and nonlinear interactions

We start with a number of examples.

x′ =iλε

x + x2, x(0) = x0. (9)

The solution of (9) can be obtained explicitly. Introducing a new variable x by

x = eiλε tw

gives us a new equation whose right hand side is bounded independent of ε

w′ = eiλε tw2, w(0) = w0 = x0.

w(t) =1

1w0

+ iελ (e

iλε t − 1)

=w0

1 + iελ w0(e

iλε t − 1)

.

As a result,

w(t) = w0

(1 − iε

λw0(e

iλε t − 1)

)+O(ε2). (10)

Thus, the nonlinear term changes the solution only by O(ε) in arbitrarily long time

w′ = f (w), f (w) =∫ 1

0eitw2dt = 0.

Hence, w(t) = w0. Indeed, we see that |w(t)− w(t)| = |w(t)−w0| ≤ C0ε for 0 ≤ t ≤T1.

An alternative solution method involves a procedure which is easier to generalize.From (9) we have,

w(t)− w0 =∫ t

0e

iλε sw2ds = −

iελ

eiλε sw2|t0 +

2iελ

∫ t

0e

iλε sww′ds

= − iελ

(e

iλε tw2(t)− w2

0

)+

2iελ

∫ t

0ei 2λ

ε sw3ds. (11)

Integrating by parts again yields an integral equation for w(t)

w(t)+iελ

eiλε tw2(t)−

4ε2

λ 2 ei 2λε w3(t) = w0 +

iελ

w20 −

4ε2

λ 2 w30 +

4ε2

λ 2

∫ t

0ei 3λ

ε sw4(s)ds.

The solution w(t) can then be constructed using fixed point iterations

w(k+1)(t) = F(w(k),w0,t)+4ε2

λ 2

∫ t

0ei 3λ

ε sw4(k)(s)ds, k = 0,1,2, · · · , (12)

averaging:

The solution is readily given by

intervals. In Sect. 3.2, we will show that w(t) is close to an effective equation from


where w(0) = w0 and

F(w,w0,t) = w0 − iελ

eiλε tw2(t)+

iελ

w20 +

4ε2

λ 2 ei 2λε w3(t)− 4ε2

λ 2 w30.

By induction, one can show that the iterations converge for t ≥ 0, 0 ≤ ε ≤ ε0 and

w(t)+iελ

eiλε tw2(t) = w0 +

iελ

w20 +O(ε2).

From (11), we have w(t) = w0 +O(ε). Hence,

w(t) = w0

(1 − iε

λw0

(e

iλε t − 1

))+O(ε2). (13)

Now, consider

x′ =iλ1

εx + y, y′ =

iλ2

εy + y2.

Changing variables to

x = eiλ1ε t u,

iλ2ε tw

yields

u′ = , w′ = eiλ2ε tw2.

w = w0 +∞

∑j=1

ε jβ jeijλ2ε t .

u = eiε (λ2−λ1)t

0 +∞

∑j=1

ε jβ jeiε (( j+1)λ2−λ1)t .

If νλ2 − λ1 6= 0 for all ν = 1,2, . . ., then

u(t) = u0 +O(ε).

However, if νλ2 = λ1, then resonance occurs and

u(t) =

u0 + εν−1β ν−1t, if ν > 1,u0 + tw0, if ν = 1.

Thus, the solution is not bounded for all time.As a generalization, consider the system

x′ =iε

Λx + P(x), x(0) = x0, (14)

where

Therefore:′ w

e i ( w1

y = e

2−λ + λ )t+ ε

From (10), we can obtain an asymptotic expansion for w:


Λ =

λ1

λ2. . .

λd

, λ1, · · · ,λd ∈ R,

and P = (p1(x), · · · , pd(x)) is a vector of polynomials in x = (x1, · · · ,xd). Let

x = eiε Λtw.

We then havew′ = e− i

ε ΛtP(

eiε Λt w

), w(0) = w0 = x0. (15)

The right hand side of (15) consists of expressions of the form

eiε (∑mjλ j)t p(w), (16)

where the m j are integers and p is a polynomial in w. There are two possibilities.

1. τ = ∑m jλ j = 0 for some terms. We call these terms the resonant modes. In thiscase (15) has the form

w′ = Q0(w)+ Qε

( tε,w)

, (17)

where Q0 contains the terms corresponding to resonant modes, and Qε the re-maining terms involving oscillatory exponentials. One can show that the solutionof

w′ = Q0(w), w(0) = w0 = x0, (18)

is very close to w for a long time; i.e.

|w(t)− w(t)| ≤ C0ε, 0 ≤ t ≤ T1, (19)

and in general w(t) does not stay close to the initial value w0.2. τ = ∑m jλ j 6= 0 for all terms. No resonance occurs in the system. The solution

stays close to the initial value:

w(t) = w0 +O(ε). (20)

We remark here that the term

f( t

ε,w)

= e− iε Λt P

(e

iε Λtw

)

is in general not strictly periodic, even though it is composed of many highly oscilla-tory terms. Nonetheless, the self averaging effect of the highly oscillatory terms canbe observed using integration by parts:


w(t) = w0 +∑τ

∫ t

0e

iτε ξ pτ(w)dξ

= w0 − iε ∑τ

1τ

eiτε ξ pτ(w)|t0 + iε ∑

τ

1τ

∫ t

0e

iτε ξ ∂ pτ

∂ pw′dξ

= w0 − iε ∑τ

1τ

eiτε ξ pτ(w)|t0 + iε ∑

τ

1τ

∫ t

0e

iτε ξ pτ(w)′dξ . (21)

The integrals in (21) are over terms of type (16) and we can therefore repeat theabove arguments. If some of the terms are not of exponential type, then they will, ingeneral, be of order O(εt). For εt 1 we can replace (21) by

w(t) = w0 − iε ∑τ

1τ

eiτε ξ pτ(w)|t0,

i.e.,w(t) = w0 +O(ε) forεt 1.

A more accurate result is

w(t) = w0 − iε ∑τ

1τ

(e

iτε ξ − 1

)pτ(y0)+ iε p0(y0)t +O(ε2t2).

If all the terms are of exponential type, then we can use integration by parts to reducethem at least to order O(ε2t). We obtain the following theorem.

Theorem 1. Assume that for all integers α j the linear combinations

∑α jλ j 6= 0.

Thenw = w0 +O(ε)

in time intervals 0 ≤ t ≤ T. T = O(ε−p) for any p.

There are no difficulties in generalizing the result and techniques to more generalequations

x′ =1ε

Λ(t)x + P(x,t).

Here Λ(t) is slowly varying and P(x,t) is a polynomial in x with slowly varyingcoefficients in time.

Remark 1. We see that without the presence of resonance, the highly oscillatory so-lution x of system (14) stays closely to

ei Λε t x0,

for a very long time. Regarding to our ultimate goal of developing efficient algo-rithms, we may conclude that if no resonance occurs in the system, no computationis needed, since ei Λ

ε tx0 is already a good approximation to the solution.


However, if resonance occurs, then the “envelop” of the oscillations in the solu-tion, x(t) changes non-trivially. In this case, efficient algorithms can be devised fromsolving the initial value problem of equation (17):

w′ = Q0(w)+ Qε

( tε,w)

, w(0) = x0.

As we showed above, one may even simply drop the oscillatory term and solve anequation that is completely independent of the fast scale:

w′ = Q0(w), w(0) = x0

and still obtain accurate approximations. We refer the readers to the paper of Scheid

side. In a later part of these notes, we shall first show that the term Qε can be easily“averaged” out without even using its explicit form. This may prove to be very usefulfor designing our multiscale algorithms for more complicated systems.

trary time intervals. Follow the steps:

1. For 0 ≤ t < T, the difference ek(t) := w(k)(t) − w(k−1)(t) converge to 0 as kapproaches infinity.

(k) is uniformly bounded for k = 1,2, . . . .3. Establish the estimate in (13).4. Arguments in the previous steps can be repeated to extend the solution to larger

time intervals.

2.2 Slowly varying solutions

Consider

εx′ = (A(t)+ εB(x,t))x + F(t), (22)

x(0) = x0,

where 0 < ε 1 and A(t), F(t) satisfy the same conditions as in the linear case, i.e.A,A−1,F and their derivatives are of order one. Also, for x of order one, B and itsderivatives with respect to x and t are of order one, and

|B| ≤ C|x|,

for some constant C > 0. Formally, taking ε = 0, yields the leading order equationAx + f = 0. Denoting the solution Φ0 = −A−1F we substitute

x = Φ0(t)+ x1,

and obtain by Taylor expansion

Exercise 2. Show that the fixed-point iterations defined in (12) converge for arbi-

[31] for an interesting algorithm that explores this special structure of the right hand

2. Show that w


εx′1 = (A(t)+ εB(x1 + Φ0,t)) (x1 + Φ0)+ F(t)− εΦ ′

0

= (A1(t)+ εB1(x1,t))x1 + εF1(t),

where B1 has the same properties as B and

A1(t) = A(t)+O(ε), F1 = B(Φ0,t)Φ0 − Φ ′0.

Thus the new system is of the same form as the original one with the forcing functionreduced to O(ε). Repeat the process p times yields

x =p−1

∑ν=0

εν Φν + xp,

εx′p = (Ap(t)+ εBp(xp,t))xp + ε pFp, Ap = A +O(ε), (23)

xp(0) = x(0)−p−1

∑ν=0

εν Φν(0).

Therefore, we have

Theorem 2. The solution of (22) has p derivatives bounded independently of ε if andonly if

x(0) =p−1

∑ν=0

εν Φν(0)+O(ε p),

i.e. x(0) is, except for terms of order O(ε p), uniquely determined.

x(0) = 0

εv′p = Ap(t)v,

is bounded, then xp = O(ε p− j) in time intervals of length O(ε− j).Generalizing (22), consider

εx′ = (A(t)+ εC(v,t)+ εB(v,x,t))x + F(t), (24)

v′ = D(v,x,t)x + G(v,t).

Here A,A−1, B,C,D,F,G and their derivatives with respect to x,v,t are of order O(1),if x,v,t are of order O(1). Following the same reasoning as before, substitute

−1(t)F(v,t)+ x1

If F and all its derivates vanish at t = 0, then the initial condition

provided we can extend F smoothly to negative t. If the solution operator of the line-

defines a solution for which any number of derivatives are bounded independent of ε .

arized problem,

x = –A

We can construct such a solution even if F and its derivatives do not vanish at t=0,


1

εx′p = (A(t)+ εCp(v,t)+ εBp(v,xp,t))xp + ε pFp(t),

v′ = Dp(v,xp,t)xp + Gp(v,t). (25)

We conclude the following:

dently of ε if we choosexp(0) = O(ε p),

i.e., x(0) is, except for terms of order O(ε p), uniquely determined by v(0).

Generalizations. More generally, we can consider systems

εw′ = h(w,t).

If there is a solution w(t) with w′(t) = O(1), then h(w(t),t) = O(ε). This suggeststhe existence of a C∞-function φ(t) with

h(φ(t),t) = 0, t ≥ 0.

Introducing the new variablew = w− φ ,

one obtains

εw′ = h(w+ φ ,t)− h(φ ,t)− εφ ′(t)= (M(t)+ N(w,t)) w− εφ ′(t)

whereM(t) = hw(φ(t),t), |N(w,t)| ≤ const. |w|.

If we further assume that

w(0) = w(0)− φ(0) = εz0, z0 = O(1),

then we can rescale the equation for w by introducing a new variable, z = ε−1w. Oneobtains for z(t)

εz′ = (M(t)+ εZ(z,t))z− φ ′(t).

Since we are interested in highly oscillatory problems, assume that M(t) has m purelyimaginary eigenvalues which are independent of t. Denote

κ j = iµ j, |µ j| ≥ δ > 0, j = 1, . . . ,m,

and n eigenvaluesκm+1 = . . . = κm+n = 0.

to obtain a system of the same form with F replaced by εF . Repeating the processp

Theorem 3. The solution of (24) has p time derivative which are bounded indepen-

times yields


Without loss of generality, assume

M =(

A 00 0

), |A−1| = O(1),

where A is an m × m matrix with eigenvalues κ j, j = 1, . . . ,m. If we partition zaccordingly,

z =(

xv

),

then we obtain

εx′ =(A + εZI(v,x,t)

)x −(φ ′)I

,

v′ = ZII(v,x,t)− 1ε(φ ′)II

.

If (φ ′)II = O(ε), then the resulting system has the form (24).

εx′ = (A(t)+ εC(v,t)+ εB(v,x,t))x,

v′ = D(v,x,t)x + G(v,t), (26)

x(0) = x0, v(0) = v0.

We obtain the slow solution vs, if we set x0 = 0, i.e.,

v′S = G(vS,t), vS(0) = v0, x ≡ 0. (27)

Let us make the following assumption.

Assumption 1. The solution operators S ,S2 of

v′L =

∂G∂v

(vS)vL

andεx′

L = (A(t)+ εC(vS,t))xL,

respectively, are uniformly bounded.

If x0 6= 0, then the slow solution will be perturbed and we want to estimate v − vS.We start with a rather crude estimate. We assume that x0

S 02

0

in time intervals 0 ≤ t ≤ T with T |x0|−1. We linearize (26) around v = vs andx = 0. Let v = vS + vL, x = xL, then the linearized equations have the form

2.3 Interaction between the fast and the slow scales

Continuing our discussion and ignoring the terms that are higher order in ε , we con-sider the following model equation:

1

|v − v | = O(|x | t + ε|x |)

Here, the solution operator S1(t, s) for t > s maps VL(s) to VL(t), and S2(t, s) acts the

is small and want to show

same way for XL.


εx′L = (A(t)+ εC(vS,t))xL,

v′L = D(vS,t)xL +

∂G∂v

(vS)vL, D = D(vS,0,t),

xL(0) = x0, vL(0) = v0.

By assumption|xL| ≤ const.|x0|.

Duhamel’s principle and integration by parts gives us

vL(t) =∫ t

0S1(t,ξ )DxLdξ

= ε∫ t

0S1(t,ξ )D(A + εC)−1x′

Ldξ

= εS1(t,ξ )D(A + εC)−1xL|t0 − ε∫ t

0

∂∂ξ

(S1(t,ξ )D(A + εC)−1)x′Ldξ .

The last integral can be treated in the same way. Therefore, Assumption 1 gives us,for any p,

|vL(t)| ≤ const.(ε|x0|+O(ε pt)) .

Assume now that A is constant, has distinct purely imaginary eigenvalues andthat B,D are polynomials in x. Our goal is to give conditions such that our estimatewill be improved to

|v − vS| = O(εx0) in time intervals 0 ≤ t ≤ T,T (ε|x0|)−1. (28)

Without restriction we can assume that the system has the simplified form

εx′ = (iΛ + εΛ1(t)+ εB(x,t))x, (29)

v′ = D(x,t)x + G(v,t), (30)

where Λ ,Λ1 are diagonal matrices and Λ1 +Λ∗1 ≤ 0. We introduce new variables

x = eiε Λt z.

Then (29) becomesz′ = Λ1(t)z+ Bz, (31)

whereB = e− i

ε ΛtB(

eiε Λt z,t

)e

iε Λt .

We splitB = B1 + B2,

where B1 is a polynomial in z without exponentials, and all terms of B2 containexponentials. B2 produces a O(ε|x0|2t)-change of z and, therefore, we neglect it.Thus, we can simplify (31) to


z′ = Λ1z+ B1(z,t)z.

If B1 6= 0, then we can in general not expect that |z| ≤ K|x0| holds in time intervalsT |x0|−1. We proved that the solution of (29) is of the form

x(t) = eiε Λt z(t), (32)

where z(t) is varying slowly. We introduce (32) into (30). We can also split

D(x,t)x = D1 + Dε , (33)

where D1 does not contain any exponentials and all terms of Dε contain exponen-tials. Observe that D1 is quadratic in z, i.e. D1 = O(|x0|2). We can further deducethat d

dt D1(x(t),t) is independent of ε , while ddt Dε ∼ O(ε−1). We have the following

important conclusion:

• If D1 6= 0, then in general

|v − vS| = O(|x0|2t).

• If D1 = 0, then|v − vS| = O(ε|x0|),

and (28) holds.

We should look at the above result together with what we obtained in Sect. 2.1, inparticular, the estimate (19) for the case when resonance occurs in the equation forx, and (20) for the case without resonance in the system.

2.4 Slow variables and slow observables

Consider the following ODE system

x′ = −ε−1y + x, x(0) = 1,y′ = ε−1x + y, y(0) = 0.

(34)

The solution of this linear system is (x(t),y(t)) = (et cosε−1t,et sinε−1t) whose

t

be decomposed into “fast and slow constituents”: a fast rotational phase and a slowlychanging amplitude. Denoting ξ = x2 2

ξ ′ =ddt

ξ (x(t),y(t)) = 2xx′ + 2yy′ = 2x2 + 2y2 = 2ξ .

Three important points call attention:

• ξ ′ is bounded independent of ε . Accordingly, we refer to the function ξ (x,y) asa slow variable for (34).

trajectory forms a slowly expanding spiral: i.e. the solution rotates around

time by e . Although both x(t) and y(t) change on the ε time scale, the system can

+ y , we have

the origin with a fast frequency 2π/ε and the distance to the origin grows in


• ξ ′ can be written into a function of the slow variable ξ only. Accordingly, we saythat the equation for the slow variable ξ is closed.

• Suppose that there is another (slow) function ζ (x,y) such that ddt ζ (x(t),y(t)) =

ζxx′ + ζyy′ is bounded independent of ε . Then, away from the origin, ∇ζ (x,y)is parallel to ∇ξ (x,y). Otherwise, at every point away from the origin, ∇ζ and∇ξ form a local basis for the two dimensional vector space. Consequently, thevelocity field Φ(x,y;ε) := (−ε−1y+ x,ε−1x+ y) can be written as a linear com-bination of ∇ζ and ∇ξ ; we write Φ = a∇ξ + b∇ζ . From the hypotheses on theslowness of ξ and ζ , Φ ·∇ξ and φ ·∇ζ are both bounded independent of ε , im-plying that the coefficients a and b are also bounded. However, this leads to Φbeing bounded which contradicts with the given equation (34).

The three observations described above are essential for building the multiscale nu-merical methods introduced in the next section.

More generally, consider the ODE systems

x′ = ε−1 f (x)+ g(x), x(0) = x0, (35)

where x ∈ Rd . We assume that for 0 < ε < ε0, and for any x0 in a region A ⊂ Rd ,

0

0 whenever it is clear from context.

Definition 1. Let U be a nonempty open subset of A . A smooth function ξ : Rd 7→ R

is said to be slow with respect to (35) in U, if there exists a constant C such that

maxx0∈U, t∈[0,T ]

∣∣∣∣ddt

ξ (x(t;ε,x0))∣∣∣∣≤ C.

Otherwise, ξ (x) is said to be fast. Similarly, we say that a quantity or constant is oforder one if it is bounded independent of ε . It is also no problem generalizing thisnotion to time dependent slow variables ξ (x,t).

Loosely speaking, ξ (x) being slow means that, to leading order in ε , the quantity

from a macroscopic domain (radius does not shrink with ε).

x = f( t

ε,x)

, f bounded,

each scalar component of the state variables x is considered a slow variable. In-

previous discussion that x(t) stays very close to its initial value for all 0 ≤ t ≤ T .Furthermore, in the case of resonance discussed in Sect. 1.3, x(t) drifts away from

1

the unique solution of (35), denoted x(t;ε ,x ), exists in t ∈ [0,T ] and stays in some

ε and xbounded region D. For brevity, we will omit the explicit dependence of the solution on

′

As another example, in Sect. 2.3 function D (y,t) may be considered as a slowvariable for (29).

deed, for those functions f that are periodic in the first argument, we know from our

Following Definition 1, for systems of the form

Hamiltonian systems, the action variables are the slow variables for the systems.the initial value in an average distance that is growing linearly in time. For integrable

ξ (x(t)) is evolving on a time scale that is ε independent for all trajectories emanating


At this point, it is natural to ask how many slow variables exist for a given highlyoscillatory system? The obvious answer is infinitely many, since any constant multi-plication of a found slow variable yield another new one. A more reasonable ques-tion is to ask what the dimension of the set of all slow variables is. In preparationto answering this question, we need to make concrete a few more concepts. In thefollowing sections this question is answered for some specific cases.

Definition 2. Let α1, · · · ,αk:A ⊂Rn 7→R be k smooth functions, k ≤ n. α1(x), . . . ,αk(x)are called functionally independent if the Jacobian has full rank; i.e.

rank

(∂ (α1, · · · ,αk)

∂x

)= k.

Let α(x) = (α1(x), . . . ,αk(x))T be a vector containing k functionally independentcomponents.

When coupled with system (35), α(x) is called a maximal vector of functionallyindependent slow variables if, for any other vector of size ν whose components arefunctionally independent, then k ≥ ν .

Our objective is to use an appropriate set of slow variables together with some othersmooth functions to provide a new coordinate system for a subset of the state spaceof system (35). Such a coordinate system separates the slow behavior from the fastoscillations and provides a way to approximate a large class of slow behavior of (35).See Fig. 4 for an illustration; locally near the trajectory, the space is decomposed intothree special directions, ∇φ defines the fast direction, and the two slow variables ξ1

and ξ2 help gauging the slow behavior of a highly oscillatory system.

ξ1

ξ2

φ

Fig. 4. Illustration of a slow chart. The slow variables ξ1 and ξ2 provide a local coordinatesystem near a trajectory.


Definition 3. Let ξ = (ξ1(x), . . . ,ξk(x)) denote a maximal vector of functionally in-dependent slow variables with respect to (35) in A , and φ : A ⊂ Rd 7→ Rd−k besome smooth functions. If the Jacobian matrix ∂ (ξ , )/∂x is nonsingular in A , oneobtains a local coordinate systems, i.e., a chart of the states space. We refer to sucha chart as a slow chart for A with respect to the ODE (35).

In other words, a slow chart is a local coordinate system in which a maximalnumber of coordinates are slow with respect to (35).

Lemma 1. Let (ξ , ) denote a slow chart for A ⊂ Rd and α(x) : A → R a slowvariable. Then, there exists a function α(ξ ) : Rk → R such that α(x) = α(ξ (x)).

Proof. Otherwise, α(x) is a new slow variable that is functionally independent of thecoordinates of ξ , in contradiction to the maximal assumption.

Another type of slow behavior can be observed through integrals of the trajectory,referred to as slow observables.

Definition 4. A bounded functional β : C1(A × [0,T ]) ∩ L1(A × [0,T ]) 7→ R iscalled a (global) slow observable if

β (t) =∫ t

0β(x(τ;ε,x0),τ)dτ.

Differentiation with respect to time shows that global observables are slow.

From the discussion in Sect. 4.2, we deduce that with an appropriate choice of kernel

β (t) =∫ +∞

∞

1η

K

(t − τ

η

)β (x(τ;ε,x0),τ)dτ,

can also be slow. We refer to these as local observables.We observe that along the trajectory passing through y0, a slow variable defines

a slow changing quantity ϑ . We first consider the unperturbed equation

εy′ = f (y,t),

and a slow variable α .

ddt

α(y(t)) = ∇α|y(t) · y′(t) =1ε

∇α|y(t) · f =: φα , f (t;y0). (36)

f 0 1

0

ε y′ = fε (y,t)+ εg(y,t).

ddt

ϑ = φα , fε (t;y0), ϑ(0) = ϑ0.

We may directly consider integrating a slow observable ϑ(t) satisfying

and η , local averages of the form

|φ0 < ε ≤ ε , then ∇α · f = 0

,α

for a neighborhood of y(t). Now consider

φ

Notice that since is a slow variable,

φ

α (t;y )| ≤ C . If this bound is valid for


Notice that

φ fε =1ε

∇α|y(t) · f (y(t),t)+ ∇α|y(t) ·g(y(t),t) = ∇α|y(t) ·g(y(t),t).

So ϑ(t) is slowly varying in O(1) time scale.

2.5 Building slow variables by parametrizing time

The time variable may be used to create slow variables that couple different oscil-lators if we can locally use the coordinates of the state space to parametrize timeso that time is treated as a dependent variable. Consider the equations of the formεy′ = f (y,t) and assume that there exists a function τ , independent of ε , such thatετ(y(t)) = t. The function τ by its definition is not slow since

ddt

τ(y(t)) =tε.

However, if we have ετ(z(t)) = t for the solutions, z(t), of another oscillatory prob-˜

ddt

θ (y(t),z(t)) ≡ 0.

hold globally. In many problems, even though the inverse function τ does not exist

integrate a slow quantity. For example, the derivative of arctan(z) is defined on thewhole real line. Similarly, on the complex plane, the derivative of the arg function

defining a continuous θ (t) on the Riemann sheet.One advantage of using time as a slow variable is in defining relative phase be-

tween two planar oscillators. Consider

εz′k = iλkzk, k = 1,2.

We formally defineα(z1,z2) := arg(z1)− arg(z2)

and obtain the equation for the slow observable. Through this approach, we can de-fine and integrate the slowly changing relative phase between two oscillators.

2.6 Effective closure

Let U(t) ∈ Rn and V (t) ∈ Rm be two smooth functions. Assume that, for all 0 ≤ t ≤T , both U(t) and V (t) are bounded above by C0 and that

The existence of inverse functions depend on the monotonicity in time of any

is defined everywhere except at the origin. In the latter case, (36) can be regarded

lem, then the function θ (y,z) := τ(y)− τ(z) is a slow variable since

coordinate of the trajectories. For oscillatory problems, the monotonicity cannot

globally, its derivative can be defined globally. In this case, we may employ (36) to


dUdt

= G(U)+ εH(U,V,t),

for some bounded smooth function H : Rn ×Rm ×R+ 7→ [−C1,C1]. We say that thedynamics of U is effectively closed. This means that for 0 ≤ t ≤ T and ε sufficientlysmall, one can ignore the influence of V (t) and compute instead

dUdt

= G(U), U(0) = U(0),

as an approximation of U(t); i.e.

|U(t)−U(t)| ≤ C1ε.

In the spiral example (34), the equation for the single slow variable ξ = x2 +y2 iseffectively closed. The following gives an example of slow variables whose dynamicsalong the trajectories are not effectively closed. In the complex plane, consider thesystem

x′ =iε

x + x∗y,

y′ =2iε

y. (37)

Here the x∗ denotes the complex conjugate of x. Evidently, ξ1 := xx∗ and ξ2 := yy∗

are two slow variables. However, the differential equation for ξ1 along the non-equilibrium trajectories of (37) is given by ξ ′

1 = 2Re((x∗)2y), which cannot be de-scribed in terms of ξ1 and ξ2 alone. Hence, the equation for ξ1 is not effectivelyclosed. In fact, it is easily verified that ξ3 = (x∗)2y is also a slow variable and that

1 2 2

Later, we will see that in many oscillatory systems, the effective equations forthe slow coordinates in a slow chart are effectively closed.

3 Averaging

One of the most important analytic tools for studying highly oscillatory systems areaveraging methods, see e.g. [5, 20, 29] . In this section we present a few key resultsand discuss them using simplified examples.

3.1 Time averaging and integration by parts

1

I(t) =∫ t

0cos( s

ε

)a(s)ds = ε sin

( tε

)a(t)− ε

∫ t

0sin( s

ε

)a′(s)ds.

function whose derivative is boun-

(ξ ,ξ ,ξ ,argx) is a slow chart.

ded on the real line. Consider the integralWe start with a simple example. Let a(t) be a C


Then |I(t)| ≤ C0(1 + t)ε for some constant C0 coming from the maximum valueof a and a′ in [0,t]. Let a(t) be a λ -periodic function in Cp, p ≥ 1; and let a∞ :=max0≤t≤λ |a(t)|. If a(t) has zero average,

∫ λ0 a(s)ds = 0, then for all T ≥ 0,

∣∣∣∣∫ T

0a(t)dt

∣∣∣∣≤ λ a∞.

We define a particular anti-derivatives of a(t) as follows

a[0](t) = a(t) and a[k](t) =∫ t

0a[k−1](s)ds+ ck, k = 1,2,3, . . . (38)

where the constant ck is chosen such that∫ λ

0 a[k](s)ds = 0. As a result, a[k](t) are alsoλ -periodic since

a[k+1](t +λ )−a[k+1](t) =∫ t+λ

ta[k](s)ds =

∫ λ

0a[k](s)ds = 0, k = 1,2,3, · · · . (39)

Consequently, all anti-derivatives are uniformly bounded:

|a[k](t)| ≤ λ a∞, ∀t. (40)

∫ T

0a( s

ε

)f (s)ds =

[εa[1]

( sε

)f (s)]T

s=0− ε

∫ T

0a[1]( s

ε

)f (1)

• If f (T ) = f (0) = 0, and f ∈ C , then∣∣∣∣∫ T

0a( s

ε

)f (s)ds

∣∣∣∣ ≤ sup0≤t≤T

| f (p)(t)|a∞ · ε p.

• If f is in C∞, we can further obtain a formal asymptotic expansion approximationfor the integral

∫ T

0a( s

ε

)f (s)ds = ∑

k

[(−1)k−1εka[k]

( sε

)f (k−1)(s)

]T

s=0.

• If a :=∫ λ

0 a(ξ )dξ 6= 0, then

Iε :=∫ T

0a( s

ε

)f (s)ds −→ I := a

(∫ T

0f (s)ds

)as ε → 0. (41)

• Similar averaging results can be obtained for functions a(t) which are not neces-sarily periodic but whose anti-derivatives are nonetheless bounded.

Exercise 3. Prove (41).

(s)ds,

p

If f is differentiable, we can perform integration by parts

where f (k) is the k-th derivative of f . The process can be repeated depending on thedifferentiability of f .


3.2 How does averaging in an oscillatory system appear?

Considerx′ = f

( tε,x)

, x(0) = x0, (42)

where f (t,x) is Lipschitz in both t and x with constant L and is λ -periodic, f (t +λ ,x) = f (t,x). In addition, consider

y′ = f (y), y(0) = y0, where f (x) =1λ

∫ λ

0f (t,x)dt. (43)

We call (43) the averaged, or effective equation derived from (42). The followingε 1

Observe that

x(t)− x0 =∫ t

0f(τ

ε,x(τ)

)dτ

=∫ t

tMf(τ

ε,x(τ)

)dτ +

M−1

∑j=0

∫ ( j+1)ελ

jελf(τ

ε,x(τ)

)dτ,

where t − tM < ελ . In each interval t j = jελ ≤ t ≤ t j+1 = ( j + 1)ελ ,

∫ ( j+1)ελf(τ

ε,x(τ)

)dτ =

∫ ( j+1)ελ

jε pf(τ

ε,x(t j)

)+O(ε)dτ

=∫ ( j+1)ελ

jελf(τ

ε,x(t j)

)dτ +O(ε2)

= ε p · 1λ

∫ λ

0f (s,x j)ds+O(ε2)

= ελ f (x j)+O(ε2).

Hence,

x(t)− x0 =∫ t

0f(τ

ε,x(τ)

)dτ

=∫ t

tMf(τ

ε,x(τ)

)dτ +

M−1

∑j=0

∫ ( j+1)ελ

jελf(τ

ε,x(τ)

)dτ

=∫ t

0f (x(τ))dτ +O(ε).

Now, since

y(t)− x0 =∫ t

0f (y(τ))dτ,

∫ t

0|x(τ)− y(τ)|dτ +Cε.

by Gronwall's lemma we have

|x(t)− y(t)| ≤ L

jελ

|x (t)− y(t)|≤ C ε for a long time which is independent of ε .calculation shows that


This result can be generalized as follows. Let

x′ =M

∑j=1

f j

( tε,x)

, x(0) = x0,

where f j is λ j-periodic in the time. Then, a calculation similar to the above showsthat the solution of

y′ = f (y), y(0) = x0,

with

f (x) = ∑j

1λ j

∫ λ j

0f j(τ,x)dτ,

is close to x(t) on a time segment.

Theorem 4. For t ∈ [0,T ], T < ∞ and independent of ε (assume x(t),y(t) exist insuch interval)

|x (t)− y(t)| ≤ C1ε.

Note that y′ = f (y) is independent of ε . While xε (t) is highly oscillatory, there areno ε-scale oscillations in y(t). We conclude that the cost of integrating the averagedequation is independent of ε and is in general much more efficient than computingxε . If we just pick an arbitrary t∗, z′ = f (t∗,z), z(0) = x0 in general we can not expectthat x(t) = z(t)+O(ε).

Averaging over oscillations may appear in many different ways and should be

which harmonic averages are derived as parameters for an effective equation.

Exercise 4. In the following problem, high frequency oscillations in aε interact withthose in d

dx uε and creates low frequency behavior of uε(x):

ddx

(aε(x) d

dx uε)

= f (x), 0 < x < 1,

uε(0) = uε(1) = 0,

aε(x) = a( xε ) > 0.

We derive an effective equation for uε(x) by performing the following steps.

1. Integrate the equation with respect to x and show that

aεduεdx =

∫ x0 f (ξ )dξ +C,

uε(x) =∫ x

0 (aε(ξ ))−1F(ξ )dξ , where F(ξ ) =∫ ξ

0 f (η)dη +C.

Determine C from boundary conditions.2. Show that

limε→0

∫ x

0a

(ξε

)−1

F(ξ )dξ =∫ 1

0a(y)−1dy

∫ x

0F(ξ )dξ , F ∈ C[0,1].

handled with caution. The following problem presents a case in homogenization, in


3. Show that

uε −→ u = A−1∫ x

0

(∫ ξ

0F(η)dη +C

)dξ as ε −→ 0,

where

A =1

∫ 10 a(y)−1dy

Ad2udx2 = f (x), 0 < x < 1, u(0) = u(1) = 0

In summary, we present the following facts about averaging, whose proof can befound, for example, in [20] and [29].

Theorem 5. Let x,y,x0 ∈ D ⊂ Rn, ε ∈ (0,ε0]. Suppose

1. f ,g, and |∇ f | are bounded by M which is independent of ε .

3. f (t,x) is λ -periodic in t, λ independent of ε .

Then, the solution of

x′ = f( t

ε,x)

+ εg( t

ε,x,ε

), x(0) = x0 (44)

is close to the solution of the averaged equation

y′ = f (x), y(0) = x0, f (y) =1λ

∫ λ

0f (t,y)dt

on a time scale of order one. More precisely, for all t ∈ [0,T ], T < ∞ independent ofε ,

|x(t)− y(t)| ≤ CεTeεLt ,

where C > 0 and L denotes a Lipschitz constant for f .

Moreover, equation (44) can be written in the form [20]

x′ = f (x)+ ε f1

( tε,x,ε

), x(0) = x0, (45)

where f1(t,x,ε) is λ -periodic in t and f1 → 0 as ε → 0.

We conclude that u(x) satisfies the effective equation:

2. g is Lipschitz in a bounded domain D.


3.3 Effective closure in coupled oscillators

Given the system (35), and in a neighborhood of the trajectory starting from x0, let(ξ ,φ) be a slow chart in which φ is a fast angular coordinate on the unit circle S1,i.e., 0 < C1/ε < φ ′

2

ξ ′ = gI(ξ ,φ),φ ′ = ε−1gII(ξ ,φ), (46)

where C1 < gII(ξ ,φ) < C2. Applying the averaging result (45), Equation (46) can berewritten as

ξ ′ =∫

gI(ξ ,φ)dφ + εgIII(ξ ,φ) = gI(ξ )+ εgIII(ξ ,φ),φ ′ = ε−1gII(ξ ,φ).

Hence, the equation for ξ is effectively closed.

4 Computational considerations

In this section we will describe a few computational methods which gain efficiencyby taking into account some of the special properties of the system discussed inprevious sections. We will mostly be concerned with equations of the form

x′ε = g

( tε,xε

), x(0) = x0,

where g(t,x) is λ -periodic, and its averaged form

x′ = g(x), x(0) = x0.

By the averaging principle, we have that

|xε(t)− ¯x(t)| ≤ Cε, 0 ≤ t ≤ T.

4.1 Stability and efficiency

Suppose uniform time stepping is used in the computations.4 The typical local trun-cation error of a p’th-order method is O((4tL)p), where L is a uniform bound forthe p + 1 derivative of the right hand side. Applied to the two equations above, theerror varies tremendously. For xε , the error term is

E1 = O

([4tε

]p),

4 With oscillatory systems, variable time step algorithms are not as advantageous in improv-ing efficiency as in stiff, dissipative systems.

< C /ε . Then, by these hypotheses, we know that


while for x, the truncation error is

E2 = O(4t p).

Thus, in order for the solution to be reasonably accurate, the step size 4t has to besmall compared to ε . In addition, typical explicit non-multiscale numerical schemessuffer from linear instabilities when using step sizes that are too large compared tothe Lipschitz constant of the right hand size. This constraint restricts the step size of

ε. On the other hand, the efficiency of solving an ODE

more efficient to solve the averaged equation for x than the original one for xε .

to construct a multiscale algorithm that solves the averaged equation without actuallyderiving it. Instead, the idea of the Heterogenous Multiscale Method is to approxi-mate the averaged equation on the fly using short time integration of the equation forxε .

4.2 Averaging kernels

In many numerical calculations involving oscillations with different frequencies, theright hand side may not be strictly periodic. As an example see (15). For this reason,as well as for efficiency considerations, it is convenient to average using some generalpurpose kernels. In the previous section, we see the need to compute the average off (t,x) over a period in t

f (x) :=1λ

∫ λ

0f (τ,x)dτ.

In this section, we show that f (x) can be accurately and efficiently approximated byaveraging with respect to a compactly supported kernel whose support is larger, butindependent of λ . For simplicity, we shall ignore the x dependence in f .

We will use Kp,q to denote the function space for kernels discussed in this paper.

Definition 5. Let K p,q(I) denote the space of normalized functions with support in I,q continuous derivatives and p vanishing moments, i.e., K ∈ Kp,q(I) if K ∈ Cq

c (R),supp(K) = I , and

∫

R

K(t)trdt =

1, r = 0;

0, 1 ≤ r ≤ p.

Furthermore, we will use Kη (t) to denote a scaling of K as

Kη (t) :=1η

K

(tη

).

For shorthand, we will also use K p,q to denote a function in Kp,q([−1,1]) .

Kexp ∈ K1,∞([−1,1]) :

such a method to be of order

In the following we will develop and discuss some of the tools and ideas required

to time T using step size 4t is O(T/4t). Hence, it is clear that it is usually much

tial kernelMost of the numerical examples in this manuscript are obtained using the exponen-


For convenience, we write f (t) = f + g(t) where

f =1λ

∫ λ

0f (s)ds.

Hence, g(t) is λ -periodic with zero average.The following analysis shows that the convolution Kη ∗ f well approximates the

¯

∫

R

1η

K

(t − s

η

)f (

sε)ds =

∫ t+η

t−η

1η

K

(t − s

η

)(f + g

( sε

))ds

= f∫ t+η

t−η

1K

(t − s

η

)ds+

1η

∫ t+η

t−ηK

(t − s

η

)g( s

ε

)ds

= f +1η

∫ t+η

t−ηK

(t − s

η

)g( s

ε

)ds.

Integrating by parts, we have

1η

∫ t+η

t−ηK

(t − s

η

)g( s

ε

)ds

=εη

K

(t − s

η

)G( s

ε

)|t+ηs=t−η − ε

η2

∫ t+η

t−ηK′(

t − sη

)G( s

ε

)ds

= − εη2

∫ t+η

t−ηK′(

t − sη

)G( s

ε

)ds,

where G is the anti-derivative of g given by (38). Hence,∣∣∣∣

1η

∫ t+η

t−ηK

(t − s

η

)g( s

ε

)ds

∣∣∣∣≤εη

||K′||∞||G||∞.

Since g is periodic and bounded, its anti-derivative is also a bounded function. Forexample, taking η =

√ε , f is approximated to order

√ε . Repeating this process q

times yields ∣∣∣∣∫

η (t − s) f (s)ds− f

∣∣∣∣≤ CK,g

(εη

)q

. (48)

f ,average

η

K

For convenience, we shall denote byKη ∗ f (t) f (t)< >η

Kexp(t) = C0χ[−1,1](t)exp(5/(t2 (47)

exp||L1(R)

Kcos(t) =12

χ[−1,1](t)(1 + cos(πt)) ∈ K1,1(I).

χ[−1,1] 0Cnormalization constant such that ||K =1. A second commonly used kernel is Here, is the characteristic function of the interval [–1, 1] and is a

−1)).


4.3 What does a multiscale algorithm approximate?

Loosely speaking, our goal is to construct an algorithm that approximates the slowbehaviour of a highly oscillatory ODE system. An important observation is that theslow behavior of a system can be a result of internal mutual cancellation of the os-cillations. This, for example, is the case with resonances. Hence, it may not be clearwhat these slow aspects are. For this reason, we take a wide approach and requirethat our algorithm approximates all variables and observables which are slow withrespect to the ODE.

How is this possible? We now prove that an algorithm which approximates theslow coordinates in a slow chart (ξ ,φ) approximates all slow variables and observ-ables.

Slow variables: Let α(x) denote a slow variable. From Lemma 1 we have thatα(x) = α(ξ (x)) for some function α . Therefore, values of α(x) depend only on ξ .Furthermore, it is not necessary to know α , for suppose ξ = ξ (x(t)) at some time t.Then, α(x(t)) = α(ξ ) = α(ξ (x(t)))). In other words, all points x which correspond

Slow observables – global time averages: We observe that for any smoothfunctions α(x,t), we have that α(t) =

∫ t0 α(x(s),s)ds is slow since |(d/dt)α(t)| =

|α(x(t),t)|, which is bounded independent of ε . In ODE form, we have

α ′ = α(x,t)

which complies to the form required by the averaging theorem. Therefore, α can beintegrated as a passive variable at the macroscopic level. In other words, it can beapproximated by

α(t) =∫ t

0< α(x(s),s) >η ds

Slow observables – local time averages: Consider time averages of the form< α(x(s)) >η . Since (ξ ,φ) is a chart, we have that α(x(s)) = α(ξ ,φ) for somefunction α . However, as proven in Sect. 4.2, convolution with kernels approximatesaveraging with respect to the fast angular phase φ . Here,

< α(x(s)) >η=< α(ξ ,φ) >η=∫

α(ξ ,φ)dφ + error = α(ξ (t))+ error,

where the error is evaluated in Sect. 4.2. Hence, a consistent approximation of ξimplies a consistent approximation of < α(x(s)) >η . Moreover, in Sect. 5 we willshow that the explicit form of α or α are not required since all local time averages

4.4 Boosting methods

In the context of averaging, the idea of boosting is particularly simple. Consider, forexample, the averaging Theorem 5 which states that, with functions f (t,x), whichare 1-periodic in time, the solution of

to the same ξ yield the same value for α(x).

can be calculated as a by product of micro-solver steps in the algorithm.

< α(x(s)) >η

dds


x′ = f( t

ε,x)

, x(0) = x0 (49)

and

y′ = f (y), y(0) = x0, f (y) =∫ 1

0f (s,y)ds, (50)

are close to order ε:|x(t)− y(t)| < Cε,

on a time scale of order one. Suppose we are interested in solving (49) with a pre-scribed accuracy ∆ which is small, but not as small than ε , i.e., ε ∆ 1. Considerthe modified equation

z′ = f( t

∆,z)

, z(0) = x0. (51)

Following the same averaging argument, z(t) is close, to order ∆ , to the averagedequation (50). Hence, by the triangle inequality

|z(t)− x(t)| ≤ |z(t)− y(t)|+ |y(t)− x(t)| < C(ε + ∆) < 2C∆ . (52)

which is of order ∆ . On the other hand, the stiffness of the equation is much reduced.The discussion in Sect. 4.1 shows that the efficiency of solving the boosted equation(51) is O(∆−1), which can be a considerable improvement over the O(ε−1) requiredto solve (49). Moreover, (51) has the exact same form as (49) and preserves thesame invariance. For example, if the original system is Hamiltonian, that the boostedversion is also Hamiltonian.

Despite their simplicity, boosting suffers from two major drawbacks. The first isrelated to the nature of the asymptotic expansion used to obtain the averaged equa-tion. Similar to expanding functions in power series, the asymptotic expansion in theaveraging Theorem 5 has a “radius of convergence”. This implies that the averagedequation may provide a poor approximation for (49) if ε is not small enough. In

0

is usually unknown. Hence, the error estimate (52) fails if ∆ > ε0.−1),

no matter what the order of the integrator is. This is not the case with HMM, as will

benchmark to test and evaluate the efficiency of our algorithm.

5 Heterogeneous Multiscale Methods

other words, the proximity between x(t) and y(t) “kicks in” at some value ε , which

Solving the boosted equation (51) instead of the original one, (49) introduces an error

Another drawback is that the efficiency of the method is bound to be O(∆

be discussed in the following section. Nonetheless, boosting serves as an important

The Heterogeneous Multiscale Method (HMM) is a general framework for systems

averaged equation, and a micro-solver, approximating the averaged equation usingHMM consists of two components: a macro-solver, integrating a generally unknown

evolving on multiple, well separated time scales. We will focus on problems with

short time integration of the full ODE system.

two time scales which are referred to as slow/fast, or macro/micro scales. An


5.1 A vanilla HMM example

Consider, for example, equations of the form

x′ = f( t

ε,x)

, x(0) = x0,

where x ∈ Rd and f (t,x) is a smooth function which is 1-periodic in t. We rewritethe system as an homogeneous equation on Rd × [0,1],

x′ = f (φ ,x), x(0) = x0,φ ′ = ε−1, s(φ) = 0,

(53)

1

in x are slow with respect to (53) while φ is fast. Hence, (x,φ) is a slow chart for(53). Furthermore, from Sect. 3, the solution for x is close (to order ε) to an effectiveequation which is effectively closed:

x′ = f (x), x(0) = x0,

where

f (z) =∫ 1

0f (τ,z)dτ.

the averaged forcing f (·) is usually unknown. For this reason, following Sect. 4.2we approximate f (·) as 〈 f (t,x(t)〉η . Applying a forward Euler scheme for x with amacroscopic step size H implies taking xn+1 = xn + H 〈 f (t,x(t)〉ηrized in the following algorithm. Let xn denote our approximation of (53) at timetn = nH.

1. n = 0n η,

tn n

3. Force evaluation: calculate Fn = 〈 xn 〉η .4. Macro-step (forward Euler example): take xn+1 = xn + HFn.5. n = n + 1. Repeat steps 2–4 to time T .

The efficiency of the algorithm is O(Tη/Hh). It is further analyzed in Sect. 5.4.

5.2 Systematically constructing heterogeneous multiscale methods

dudt

= fε (u,t), (54)

where u : (0,T ) 7→ Rd , and a subset of the eigenvalues of ∂ fε/∂u are inversely pro-portional to a small positive parameter ε . When ε is very small, the complexity of

Earlier we saw that it is much favorable to solve for x rather than for x. However,

is isomorphic to the unit circle S . By Definition 1, it is clear that all the coordinates

. This is summa-

Consider stiff ordinary differential equations (ODEs) of the form

Denote the solution x (t).+ η ] with step size h and xn0x replaced byf .

ε( ),, (·)

where φ is an angular variable defined on the quotient space R/[0,1]. The latter space

2. Micro-simulation: approximate (53) numerically in a reduced time segment [t −


numerically solving such systems becomes prohibitively high. However, in manysituations, one is interested only in a set of quantities U that are derived from thesolution of the given stiff system (54), and typically, these quantities change slowlyin time; i.e. both U and dU/dt are bounded independent of ε . For example, U couldbe the averaged kinetic energy of a particle system u.

Our objective is to construct and analyze ODE solvers that integrate the system

ddt

U = F(U,D), (55)

where D is the data that can be computed by local solution of (54). U is calledthe slow (macroscopic) variable that is also some function or functional of u; i.e.U = U(u,t).

strategy involves setting up a formal numerical discretization for (55), and evaluatesF from short time history of u with properly chosen initial condition.

Φ(U,D) = 0, (M) (56)

which may not be explicitly given, but can be evaluated from a given microscopicmodel,

ϕ(u,d) = 0, u ∈ Ω(m) (57)

where u are the microscopic variables. D = D(u) and d = d(U) denote the set ofdata or auxiliary conditions that further couple the macro- and microscopic models.Model (56) is formally discretized at a macroscopic scale, and the adopted numericalscheme dictates when the necessary information D(u) should be acquired from solv-ing (57), locally on the microscopic scale with auxiliary conditions d(U). As part ofd(U) and D(u), the macro- and microscopic variables are related by reconstructionand compression operators:

R(U,DR) = u, Q(u) = U, Q(R(U,DR)) = U,

where DR are the needed data that can be evaluated from u. Errors of this type ofschemes generally take the structure [10, 14]

Error = EH + Eh,

where EH is the error of the macroscopic model (56), and Eh contains the errorsfrom solving (57) and the passing of information through R and Q. This approachhas been used in a number of applications, such as contact line problems, epitaxialgrowth, thermal expansions, and combustion. See the review article [12].

Figure 1 shows two typical structures of such ODE solvers. An ODE solver for Ulies on the upper axis and constructs approximations of U at the grid points depicted

If F is well-defined and has a convenient explicit mathematical expression, then

U ∈ Ω

many situations, the dependence of F on U is not explicitly available. Our proposed

multiscale methods. In this framework, one assumes a macroscopic model

there is no need to solve the stiff system (5 4) — one only needs to solve (55). In

We will follow the framework of E and Engquist [11] in constructing efficient


there. The fine meshes on the lower axis depict the very short evolutions of (54)with initial values determined by R(U(tn)). The reconstruction operator then takeseach time evolution of u and evaluates F and U . The algorithms in [10, 16, 15], and[28] are also of a similar structure. As a simple example, the forward Euler schemeapplied to (55) would appear to be

Un+1 = Un + H · F(Un), (58)

where F contains the passage of QΦtR(Un) — reconstruction R, evolution Φt , and

• With the system for u, and a choice of U(u), is F well-defined by the proceduredefined above? If not, how can it be properly defined?

• What are R and Q?• How long should each evolution be computed?• What does consistency mean?• What about stability and convergence?

For a fixed given ε > 0, all well known methods will converge as the step-size H → 0,and there is no difference between stiff and non-stiff problems. In [11], convergencefor stiff problems (ε H) is defined by the following error:

E(H) = max0≤tn≤T

( sup0<ε<ε0(H)

|U(tn)−Un|). (59)

Here, ε0(H) is a positive function of H, serving as an upper bound for the range of ε ,and U(tn) and Un denote respectively the analytical solution and the discrete solutionat tnvarying property of U and generate accurate approximation with a complexity that issublinear in ε−1.

The problems we are interested in can be described as follows. A full scale sys-tem (54) written in the unknown variable u is given, and the oscillations in u havefrequency of order ε−1

assumed that the fine scale system describes the full behavior of the problem. Wewant to compute the effective behavior of the given full scale system using a number

functionals of u, and their governing equations may have noexplicit analytical for

use numerical solutions of u to extract the missing informationneeded to evaluate the formal discretization of the governing equations.

Notation 1. Let u(t;α) denote the solution of the initial value problem:

dudt

= fε (u,t), u(t∗) = α, (60)

Essential questions that need to be resolved for such a scheme include:

mula. Our approach is to discretize the effective equations for(U,V ) formally and

of slowly changing quantities, (U,V ). These slowly changing quantities generallydefined as functions or

compression Q, and H is the step size. If each evolution of the full scale system (54)is reasonably short, the overall complexity of such type of solvers would be smaller

R5.1 uses the identity operator for both and .Q

than solving the stiff system (54) for all time. The vanilla HMM presented in Sect.

= nH. With this notion, it is clear that a sensible method has to utilize the slowly

. We shall also call this system the fine scale system. It is


for some arbitrary initial condition at time t∗.The notation u(t) or u will be reserved for the solution of the same ODE, Equa-

tion (60), for t > 0 with the given initial condition u0.

Definition 6. Let G (·,t) be a functional of u. The initial data α is said to be consis-tent with u under G if G (u( · ;α),t) = G (u,t)+O(εr), for some r > 0.

In Kapitza’s pendulum problem, the pivot of a rigid pendulum with length l is at-tached to a strong periodic forcing, vibrating vertically with period ε . The systemhas one degree of freedom, and can be described by the angle, θ , between the pen-dulum arm and the upward vertical direction:

lθ ′′ =(

g +1ε

sin(

2πtε

))sin(θ ), (61)

with initial conditions θ (0) = θ0,θ ′(0) = ω0. With large ε , the only stable equilibriaare θ0 = nπ , corresponding to the pendulum pointing downward. When ε is suffi-ciently small and both θ0 and ω0 are close to 0, the pendulum will oscillate slowlyback and forth, pointing upward, with displacement θ < θmax. The set up of thependulum and an example solution are depicted in Fig. 5. The period of the slowoscillation is, to leading order in ε , bounded independent of the forcing period ε .On top of the slow motion around the stable θ = 0 configuration, the trajectory of θexhibits fast oscillations with amplitude and period proportional to ε .

In [32], the second order equation is written as a first order system using u =(θ ,ω), where ω is the derivative of θ . The slow variable U = (Θ ,Ω) consists ofthe weak limit of the angle θ and its derivative θ , and the effective force for U canbe adequately approximated by the time averaging of the right hand side of (61).However, the reconstruction operator R can no longer be the identity operator. Theinitial values of u at tn for each fine scale evolution should be carefully constructedsuch that the averages of θ matches with Θ in order to keep the correct resonancebetween the terms sin(2πt/ε) and sin(θ ). To this end, the reconstruction operatormust carry a correction term when setting up ω at tn :

ω0n = Ωn − 1

ε

∫ tn+ε/2

tn−ε/2

∫ t

tnaε

( sε

)sin(θn(s))dsdt.

Consistency of the described multiscale solver to this type of system is thus estab-lished.

d ×Rs, whose

the solution U(t) of the ODE

ddt

U =ddt

(VW

)= F (U,t)+O(ε), (62)

is equivalent to its evaluation using the whole scale solution u, i.e,

7

Problem 1 (Closure). Given V which consists of a set of slow variables or functionalsU = (V,W ) : [0,T ] → Rof u, determine the set of extended variables

dent of such thatcomponents are functions or functionals of u, so that there exists a function F indepen-

ε


Fig. 5. (a) Kapitza’s pendulum; (b) The slow scale solutions to equation (61); Three ordersof magnitude (ε = 10−6) separate the period of the slow oscillation apparent in the graphsfrom the fast oscillation.

U(t) = U(u,t) =(

V (u,t)W (u,t)

),

where U(t) denotes the ODE solution, and U(u,t) the functional evaluation usingu(t), and similarly for V and W .

When these functions are composed of u(t) and viewed as functions of time,αε (t) = α(u(t)) and ψε(t) = ψ(u(t)), they should satisfy the following conditions:

1. α and ψ are linear combinations of some simple functions of u;2. dαε/dt is bounded independent of ε;3. dν 〈ψε〉/dtν > > 0 for some 0 ≤ ν and for some independent of ε;4. αε (t) converges pointwise to a smooth function α(t), and ψε weakly to a con-

tinuous function Ψ .

Here, 〈ψε〉 denotes a moving average with respect to a kernel, as described inSect.4.2. These approaches are motivated by the analysis of resonance, the averagingmethods, see e.g. [25, 5, 1, 3], and our previous work on Kapitza’s pendulum and afew other model problems. Another interesting point of view makes use of the ideaof Young measures [4].

In practice, we do not have u(t), since we do not solve the stiff equation for along time interval independent of ε . However, the solution U to the closure problemdefines an equivalence class for the initial conditions for u. As long as an initial datais selected such that it is consistent to u(t) with respect to U(t), U(t) is properlyevolved. Instead, our strategy is to compute the solution u(·;a) for a duration thatvanishes with ε , starting from a specified time and using some initial values a. OnceU(t) is approximated, we can approximate dU/dt numerically without explicitly

W (u,t).One of our strategies is to look for algebraic functions α and ψ when constructing

δ δ

(a)

0 5 10 15

−0.4

−0.2

0

0.2

0.4

0.6

t

(b)


to the functional U(u,t):

Un at tn that specifies a set of constraints.

Find an ∈ Rd such that U(u(·;an),tn) = Un +O(ε p) for some p > 0.

Note that unlike the common constraints of conserved integrals in the computations

j) isknown at t = t j,

1. Find a j that solves U(u(·;a j),t ) ' U(t j) (Reinitialization);2. Solve the given stiff equation and obtain u(t;a j) for t j ≤ t ≤ t j +ηε (Microscale

solution);3. Evaluate F (t j) = dU/dt at t j using u(t;a j);4. Use F (t j) and U(t j) to get U at t j + ∆ t. (Macroscale solution)

Note that ∆ t should be independent of ε and ηε vanish with ε.In the following examples the slow behavior is approximated using functions

only (the slow chart) and not functionals.

5.3 Example: an expanding spiral

Consider the system (34) describing the expanding spiral

x′ = −ε−1y + x, x(0) = 1,y′ = ε−1x + y, y(0) = 0.

(63)

Previously, in Sect. 2.4, it was shown that (ξ ,φ) = (x2 + y2, tan−1(y/x)) is a slowchart for (63). The time evolution of the only slow variable ξ takes the form

ξ ′ =⟨ξ ′⟩

η +O(ε) =⟨2xx′ + 2yy′⟩

η . (64)

This motivates the following multiscale algorithm for approximating ξ ′(t). For sim-

We denote tn = nH and byxn, yn and ξn our n n n n

and yn do not have n n

The algorithm is depicted in Fig. 6.

1. Initial conditions: (x(0),y(0)) = (x0,y0), n = 0.2. Micro-simulation: Solve (63) in [tn − η/2,tn + η/2] with initial conditions

(x(tn),y(tn)) = (xn,yn).

evaluating F . Naturally, this initial value α should be consistent with u with respect

by thecomponents of U , that can be slowly varying in time.

Problem 2 (Reintialization/reconstruction). Given a functionalU(u,t) and its value

In summary, our multiscale method is outlined as follow: Assuming U(t

j

sistent and stable integrator can be used as micro-solver.plicity, we apply a macroscopic forward Euler solver with step size H. Any con-

approximation for x(t ), y(t ) and ξ (t ), respectively. Note that xto be close to x(t ) and y(t ). We only require that the slow vari-

able is approximated.

of Hamiltonian systems, we consider constraints, such as those specified


micro−solver

micro−solver

H

H

x1

x2

Fig. 6. Two macroscopic steps for the HMM algorithm in the expanding spiral example (63).

3. Force estimation: Approximate ξ ′ by ∆ξn = 〈2xx′ + 2yy′〉η . The step involvesconvoluting 2xx′ + 2yy′ with an averaging kernel as discussed in Sect. 4.2.

4. Macro-step (forward Euler): ξn+1 = ξn + H∆ξn.5. Reconstruction (second order accurate): (xn+1,yn+1) = (xn,yn)+HFn, where Fn

is the least squares solution of the linear system

Fn ·∇ξ (xn,yn) = ∆ξn

6. n = n + 1. Repeat steps 2–5 to time T .

5.4 HMM using slow charts

Suppose an ODE system of the form (35) admits a slow chart (ξ ,φ), where ξ =(ξ 1, . . . ,ξ k) ∈ Rk are slow and φ ∈ S1 is fast. In the next section we will see thatmany highly oscillatory systems indeed admit a slow chart of that form. Then, thealgorithm suggested in the previous section can be easily generalized as follows.As before, for simplicity we concentrate on the forward Euler case. Higher ordermethods are considered in [1]. Approximated quantities at the n‘th macroscopic timestep are denoted by a subscript n.

1. Initial conditions: x(0) = x0, n = 0.2. Micro-simulation: Solve (35) in [tn − η/2,tn + η/2] with initial conditions

x(tn) = xn

3. Force estimation: Approximate ξ ′ by ∆ξn = 〈∇ξ · x′〉η using convolution withan averaging kernel.

4. Macro-step (forward Euler): ξn+1 = ξn + H∆ξn.


5. Reconstruction (second order accurate): xn+1 = xn + HFn, where Fn is the leastsquares solution of the linear system

Fn ·∇ξ (xn) = ∆ξn. (65)

6. n = n + 1. Repeat steps 2–5 to time T .

Complexity

In this section we analyze the accuracy of the suggested method outlined above.Each step of the approximations preformed in our algorithm introduces a numericalerror. In order to optimize performance, the different sources of errors are balancedto a fixed prescribed accuracy ∆ . We show how the different parameters: ε , η , hand H scale with ∆ in order to have a global accuracy of order ∆ . Note that themaximal possible accuracy is ∆ = ε , since this is the error introduced by simulatingthe averaged equation rather than the original one. We also study the ∆ dependenceof the complexity of the algorithm.

We begin with estimating the error in our evaluation of the averaged force ∆ξn.There are several sources of errors:

• Global error in each micro-simulation. Using an m’th order method with step sizeh the global error is ηhm/εm+1.

• Quadrature error in K′η ∗ ξ : Using a quadrature formula of degree r the error is

ηhm/ε(m + 1). However, due to the regularity of the kernel used K ∈ Cq, theintegrand is smooth and periodic. Hence, the coefficients of its Fourier decompo-sition decay very fast. As a result, it is advantageous to use the trapezoidal rule,which is exact for e2π ikx, k ∈ N. This implies that the quadrature error is typicallyvery small and can be neglected.

• Approximating ∆ξn by 〈∇ξ · x′〉η : Using a kernel K ∈ Kp,q the error is the largerbetween η p q

(48) since ∆ξn is found through integration by parts (cf. Sect. 5.4). The above twobounds to the averaging error are equal if η p+q+1 = εq, where, for large η , theterm η p dominates, while for small η the other. Since we would like to optimizeour complexity, it is always preferable to work in the latter regime. Hence, wecan take the averaging error to be (ε/η)q/η .

Balancing all terms yields the optimal scaling of the simulation parameters with ∆ .The global accuracy of integrating the original full ODE to time T = O(1) using

a macro-solver of order s with step size H is, assuming errors are accumulative,

E ≤ Dmax

Hs,

ηhm

εm+1 ,εq

ηq+1

, (66)

For some D > 0. For short hand we drop the constant in all following expressions.Balancing the different sources of errors to a prescribed accuracy ∆ yields

and (ε/η) /η . Note that we are losing one order of η compared to


η = εq

q+1 ∆− 1q+1 ,

H = ∆1s ,

h = ε1+ 1m(q+1) ∆

s+1sm + 1

m(q+1) .

(67)

The complexity is then

C =ηh

TH

= ε− m+1m(q+1) ∆− 1

s − s+1sm − m+1

m(q+1) . (68)

With a smooth kernel we can consider the q → ∞ limit. In this case the complexityestimate is reduced to

C(q → ∞) = ∆− 1s − s+1

sm . (69)

Figure 7 depicts the relative error of the HMM approximation compared to theanalytical solution of the expanding spiral example (34). The kernel was constructedfrom polynomials to have exactly two continuous derivatives and a single vanish-ing moments, i.e., q = 2 and p = 1. Fourth order Runge–Kutta schemes were usedfor both the micro- and the macro-solvers. The simulation parameters are chosen tobalance all errors as discussed above.

10−3

10−2

10−1

10−1

100

101

102

ln(∆)

ln(r

elat

ive

erro

r in

%)

slope=1.02

Fig. 7. A log-log plot of the relative error of the HMM approximation to a linear ODE com-pared to the exact solution: E = maxtn∈[0,T ] 100 × |ξHMM(tn) − ξexact(tn)|/|ξexact(tn)|, as afunction of ∆ .

From the parameter scaling (67) it is clear that the step size of the macro-solver,H, does not depend on the stiffness ε , but only on the prescribed accuracy ∆ . Our

0

[11]. More precisely, denote the sample times of the macro-solver by t0 = 0, . . . ,tN =T and the corresponding numerical approximations for x by x0, . . . ,xN . The exactsolution is denoted x(t). We have that, for any variable α(x) that is slow with respectto x(t)

limH→0

supk=0,...,N

supε<ε0

|α(x(tk))− α(xk)| → 0. (70)

algorithm is therefore multiscale is the sense that it converges uniformly for all ε < ε


Note that the order of the limits is important.

5.5 Almost linear oscillators in resonance

Consider an ODE system of the form

x′ =1ε

Ax + f (x), (71)

where, x ∈ Rd

1 r

i j and ni j, such that mi jωi =ni jω j.

Theorem 6. There exists a slow chart (ξ ,φ) in Rd \ ξi 6= 0,∀i for (71) such thatall the coordinates of ξ are polynomial in x and φ ∈ S1.

The theorem is proven in [1]. As an example, consider (71) with

Changing variables so that A is diagonalized yields the complex system

where z = (z1,z∗1,z2,z∗

2)T and z∗ denotes the complex conjugate of z. It is easily

verified that the following are slow variables

ξ1 = z1z∗1,

ξ2 = z2z∗2,

ξ3 = z21z∗

2.

Transforming back to the original coordinates x = (x1,v1,x2,v2) the slow variablesbecome the real polynomials

ξ1 = x21 + v2

1,ξ2 = x2

2 + v22,

ξ3 = x1x22 + 2v1x2v2 − x1v2

2.

1 and ξ2 correspond to the square of the amplitude of the1 1 2 2

ξ3, corresponds to the relative propagation of phase in the two oscillators.(x2,v2) increases twice as fast

1 1

ξThe first two variables,(x ,v ) and (x ,v ), respectively. The thirdtwo harmonic oscillators described by

variable,It is slow because, to leading order in ε , the phase of

(x ,v ).as that of

z0 = z + ~f(x):

and A is an d × d real diagonalizable matrix with purely imaginaryeigenvalues ±iω , . . . ,±iω , 2r = d. In addition, we assume that all oscillatorymodes are in resonance. This implies that the ratio of every pair of frequencies is rational, i.e., for all i, j = 1 . . .r, there exist integers m

A =

1−1

2−2

.

0 0 00 0 0

0 0 00 0 0

1ε

i−i

2i−2i

0 0 00 0 00 0 00 0 0


Exercise 5. Verify that ∇ξ1, ∇ξ2, and ∇ξ3 are not linearly dependent in any regionin R4 \ Q where Q is the zeros of ξ1, ξ2, and ξ3.

Fully nonlinear oscillators

Dealing with non-linear oscillators is more complicated than linear ones. However,the slow behavior of weakly coupled systems of oscillators such as Van der Pol,relaxation and Volterra-Lotka can still be described using some generalization ofamplitude and relative phase. This is beyond the scope of these notes. We refer to [2]for further reading.

6 Computational exercises

Computer exercise 1. Let u = (x,y,z) and

fε (x,y,z) =

a 1

ε 0− 1

ε b 00 0 − 1

10

xyz

+

00

x2 + cy2

. (72)

The equation for u isu′ = fε (u), u(0) = (1,0,1).

Take ε = 10−4, a = b = 0 and c = 1. Find approximations for z(t) in 0 < t ≤ 1 usingthe following schemes and compare with the analytical solution. Plot the trajectoriesof your approximations of x(t) and y(t) on the xy-plane, and the graph z(t) as afunction of time. Explain what you observe in each case.

(a) Forward Euler using 4t = ε/50.(b) Backward Euler for x and y and Forward Euler for z, using 4t = 0.1.(c) Verlet method or Midpoint rule for x and y, and Forward Euler for z, using 4t =

ε/50.(d) Solve this problem by the HMM–FE–fe method (see below), with Q = R = I

(see Sect. 5.2). h = ε/50, H = 0.1, and hM = 2 ·10−3.(e) Derive linear stability criteria on H for HMM–FE–fe, assuming that h = c0ε .(f) Let a = b = 1 in the system defined above. Solve it by the same HMM–FE–fe

scheme with the same parameters as in (d). Does this scheme correctly approxi-mate the behavior of z in the time interval 0 < t ≤ 1? Explain.

HMM–FE–fe scheme for u′ = fε (u).

• Macroscale with Forward Euler (FE)

Un+1 = Un + HFn, U0 = Q(u0)


• Microscale with Forward Euler (fe)

unk+1 = un

k + h fε(unk), k = 0,±1, · · · ,±M,

un0 = R(Un).

• Averaging

Fn :=1

2M

M

∑k=−M

Kcos(

k2M

)fε (un

k),

Kcos(t) =12

χ[−1,1](t)(1 + cos(πt)) ,

χ[−1,1](x) =

1, −1 ≤ x ≤ 1,

0, otherwise.

Computer exercise 2. Following the previous problem, define the slow variable

ξ (x,y) = x2 + y2 and ξ (t) := x2(t)+ y2(t),

where x(t) and y(t) are defined in (72).

(a) Show that dξ/dt can be approximated by averaging:∣∣∣∣dξdt

(tn)−∫ ∞

−∞− d

dtKcos

(tn − t2Mh

)(x2(t)+ y2(t)

)dt

∣∣∣∣≤ Cη p.

Find p.(b) Modify your previous HMM–FE–fe code to HMM–FE–rk4 (see below) as fol-

lows and determine if the dynamics of z is accurately approximated by this newscheme. Plot your approximations as in the previous problem. Explain your find-ings.

(c) Do the same thing as in the previous problem, but with c = 0. Does your multi-scale algorithm work? Why?

Constrained HMM–FE–rk4 scheme for u′ = fε (u).

• Macroscale with Forward Euler

Un+1 = Un + HFn, U0 = Q(u0).

• Microscale with Runge–Kutta 4 (rk4)

unk+1 = rk4(un

k ,h), k = 0,±1, · · · ,±M,

un0 = R(Un).

Here rk4 is an explicit Runge–Kutta 4 routine using step size h.

rk4(y,h) = y +16(k1 + 2k2 + 2k3 + k4),

k1 = h fε(y), k2 = h fε(y +12

k1), k3 = h fε(y +12

k2), k4 = h fε (y + k3).


• Averaging

dzn :=1

2M

M

∑k=−M

Kcos(

k2M

)(xn

k · xnk + cyn

k · ynk −

znk

10

).

dξ n :=1

2M

M

∑k=−M

G

(k

2M

)(xn

k · xnk + yn

k · ynk) ,

where G( k2M ) := −1

2Mhddt Kcos( t

2Mh ).• Evaluate effective force

Find a unit vector dXn such that

dξ n = ∇x,yξ |xnk ,yn

k·dXn.

Fn :=(

dXn

dzn

).

Computer exercise 3. Consider the inverted pendulum equation:

lθ ′′ =(

g +1ε

sin(

2πtε

))sin(θ ). (73)

Let ω = θ ′, rewrite it into a system of first order equations for (θ ,ω). Let Ωn+ 12

denote the averaged macroscopic angular momentum at time (n + 12 )H and Θn be

the averaged macroscopic angle. Compute the inverted pendulum solutions by usingthe parameters ε = 10−6, (Θ0,Ω0) = (0.0,−0.4), g = 0.1, l = 0.05. Experiment withη = 10ε and 30ε.This problem is analyzed in [32].

HMM for the inverted pendulum problem.

• Macroscale with VerletGiven Un = (Θ n,Ω n), for n = 0,1,2, ldots

Ω n+ 12 = Ω n +

H2

· Fn,

Θ n+1 = Θ n + H ·Ω n+ 12 ,

Ω n+1 = Ω n+ 12 +

H2

· Fn+1,

Here, F[θ n,ωn] denotes the averaged force using the solutions whose values attn = nH are (θ n,ωn).

• Microscale evolutionSolve lθ ′′ = (g+ 1

ε sin(2π tε ))sin(θ ) for tn−η ≤ t ≤ tn+η with the “reconstructed

initial”


θ (tn) = Θ n,

ω(tn) = θ ′(tn) = R(Θ n,Ω n) := Ω n − sin(Θ n)cos(2π tn

ε)

2π l.

(ω(tn) ≈ Ω n −

⟨∫ t

tnaε

( sε

)sin(θ (s))ds

⟩)

• AverageUsing the solution computed in the microscale evolutions around tn. Evaluate

Fn =⟨

(g +1ε

sin(

2πtε

))sin(θ )

⟩

η= Kη ∗ f (tn),

where

Kη (t) =422.11

ηexp

[5

(4t2

η2 − 1

)−1]

,

and

f (t) =(

g +1ε

sin(

2πtε

))sin(θ (t)).

Use the Trapezoidal rule to approximate the above convolutions.

Computer exercise 4. The following is a well studied system taken from the theoryof stellar orbits in a galaxy

r′′1 + a2r1 = εr2

2r′′

2 + b2r2 = 2εr1r2.

Rewrite the above equation into the standard form (35).

(a) To see how resonances occur, change into polar coordinates and take a = ±2b.(b) Let a = 2 and b = 1. Find a maximal slow chart.(c) Apply the HMM algorithm described in Sect. 5.4 to approximate the slow be-

haviour of the system.

References

1. G. Ariel, B. Engquist, and Y.-H. Tsai. A multiscale method for highly oscillatory ordinarydifferential equations with resonance. Math. Comp., 2008. To appear.

2. G. Ariel, B. Engquist, and Y.-H. Tsai. Numerical multiscale methods for coupled oscilla-

3. V.I. Arnol’d. Mathematical methods of classical mechanics. New York, Springer-Verlag,2 edition, 1989.

4. Z Artstein, J. Linshiz, and E.S. Titi. Young measure approach to computing slowly ad-vancing fast oscillations. Multiscale Model. Simul., 6(4):1085–1097, 2007.

5. N. N. Bogoliubov and Yu. A. Mitropolski. Asymptotic Methods in the Theory of NonlinearOscillations. Gordon and Breach, New York, 1961.

6. R. Car and M. Parrinello. Unified approach for molecular dynamics and density functionaltheory. Phys. Rev. Lett., 55(22):2471–2475, 1985.

tors. Multiscale Model. Simul., 2008. Accepted.


7. A.J. Chorin. A numerical method for solving incompressible viscous flow problems. J.Comp. Phys., 2:12–26, 1967.

8. G. Dahlquist. A special stability problem for linear multistep methods. Nordisk Tidskr.Informations-Behandling, 3:27–43, 1963.

9. G. Dahlquist, L. Edsberg, G. Skollermo, and G. Soderlind. Are the numerical methodsand software satisfactory for chemical kinetics? In Numerical Integration of DifferentialEquations and Large Linear Systems, volume 968 of Lecture Notes in Math., pages 149–164. Springer-Verlag, 1982.

10. W. E. Analysis of the heterogeneous multiscale method for ordinary differential equa-tions. Commun. Math. Sci., 1(3):423–436, 2003.

11. W. E and B. Engquist. The heterogeneous multiscale methods. Commun. Math. Sci.,1(1):87–132, 2003.

12. W. E, B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden. Heterogeneous multiscalemethods: A review. Commun. Comput. Phys., 2(3):367–450, 2007.

13. B. Engquist and O. Runborg. Computational high frequency wave propagation. ActaNumerica, 12:181–266, 2003.

14. B. Engquist and Y.-H. Tsai. Heterogeneous multiscale methods for stiff ordinary differ-ential equations. Math. Comp., 74(252):1707–1742, 2003.

15. C. W. Gear and I. G. Kevrekidis. Projective methods for stiff differential equations: prob-lems with gaps in their eigenvalue spectrum. SIAM J. Sci. Comput., 24(4):1091–1106(electronic), 2003.

16. C. W. Gear and I. G. Kevrekidis. Constraint-defined manifolds: a legacy code approachto low-dimensional computation. J. Sci. Comput., 25(1-2):17–28, 2005.

17. D. Ter Haar, editor. Collected Papers of P.L. Kapitza, volume II. Pergamon Press, 1965.18. E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration, volume 31 of

Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations.

19. E. Hairer and G. Wanner. Solving ordinary differential equations. II, volume 14 ofSpringer Series in Computational Mathematics. Springer-Verlag, 1996.

20. J. Hale. Ordinary differential equations. New York, Wiley-Interscience, 1969.21. J. B. Keller. Geometrical theory of diffraction. J. Opt. Soc. Amer., 52:116–130, 1962.22. J. Kevorkian and J. D. Cole. Perturbation Methods in Applied Mathematics, volume 34 of

Applied Mathematical Sciences. Springer-Verlag, New York, Heidelberg, Berlin, 1980.23. J. Kevorkian and J. D. Cole. Multiple Scale and Singular Perturbation Methods, volume

114 of Applied Mathematical Sciences. Springer-Verlag, New York, Berlin, Heidelberg,1996.

24. H.-O. Kreiss. Problems with different time scales. Acta Numerica, 1:101–139, 1992.25. H.-O. Kreiss and J. Lorenz. Manifolds of slow solutions for highly oscillatory problems.

Indiana University Mathematics Journal, 42(4):1169–1191, 1993.26. B. Leimkuhler and S. Reich. Simulating Hamiltonian dynamics, volume 14 of Cambridge

Monographs on Applied and Computational Mathematics. Cambridge University Press,2004.

27. L. R. Petzold. An efficient numerical method for highly oscillatory ordinary differentialequations. SIAM J. Numer. Anal., 18(3):455–479, 2003.

28. L. R. Petzold, O. J. Laurent, and Y. Jeng. Numerical solution of highly oscillatory ordinarydifferential equations. Acta Numerica, 6:437–483, 1997.

29. J. A. Sanders and F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems,volume 59 of Applied Mathematical Sciences. Springer-Verlag, New York, Berlin, Hei-delberg, Tokyo, 1985.


30. R. E. Scheid. The accurate numerical solution of highly oscillatory ordinary differentialequations. Math. Comp., 41(164):487–509, 1983.

31. R. E. Scheid. Difference methods for problems with different time scales. Math. Comp.,44(169):81–92, 1985.

32. R. Sharp, Y.-H. Tsai, and B. Engquist. Multiple time scale numerical methods for theinverted pendulum problem. In B. Engquist, P. Lotstedt, and O. Runborg, editors, Multi-scale methods in science and engineering, volume 44 of Lect. Notes Comput. Sci. Eng.,pages 241–261. Springer, Berlin, 2005.

33. E. Vanden-Eijnden. On HMM-like integrators and projective integration methods forsystems with multiple time scales. Commun. Math. Sci., 5(2):495–505, 2007.

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Multiscale Computations for Highly Oscillatory Problems